On Prediction Interval for Independent Observations

Home , Prediction interval

Applied Mathematical Sciences, Vol. 9, 2015, no. 99, 4931 - 4940 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.54366

Khreshna Syuhada

Statistics Research Division, Institut Teknologi Bandung Jalan Ganesa 10 Bandung, Indonesia

Rizky Saputra

Statistics Research Division, Institut Teknologi Bandung Jalan Ganesa 10 Bandung, Indonesia

Copyright c 2015 Khreshna Syuhada and Rizky Saputra. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We consider the problem of prediction interval for future observations in the case of normal random variables. Specifically, two prediction intervals are developed with the difference lies on parameter estimators. The resulting estimative prediction intervals have coverage probability as well as expected length bounded to O(n−1). Furthermore, whilst the coverage probability 1 − α is usually given, we propose to set this as parameter. Consequently, the O(n−1) terms in the asymptotic expansion of coverage probability and of expected length will depend not only on model parameter but also α. Our aim is to find an optimal coverage probability. Numerical analysis is carried out to illustrate the unexpected coverage probability and length of estimative prediction interval. Keywords: expected length, estimative prediction interval, normal distribution, optimal coverage

1 Introduction

The accuracy of prediction interval may be assessed by calculating its coverage probability. Alternatively, we can assess prediction interval via expected 4932 Khreshna Syuhada and Rizky Saputra length, see e.g. Kabaila and Syuhada (2007) who derived the relative eﬃciency of prediction intervals by measuring the ratio of their expected length.

Suppose that Y1,Y2,...,Yn,Yn+1 are random variables that are independent and identically distributed with parameter vector θ. The available data are Y1,Y2,...,Yn and we wish to find a prediction interval, I(θ), for Z = Yn+1 such that P Z ∈ I(θ) = 1 − α, for all θ and α is level of significance. For unknown θ, an estimative prediction interval I(Θ)b is developed by re- placing a specified estimator Θb to θ. Consequently, the coverage probability of I(Θ)b is 1 − α + O(n−1) due to parameter variability. The expected length of estimative prediction interval, E I(Θ)b , is also bounded to O(n−1). Kabaila and Syuhada (2007, 2010) argued that there can be a trade off be- tween O(n−1) terms in the asymptotic expansions of coverage probability and of expected length. To avoid this trade off and make use of expected length, an improved prediction interval I+(Θ)b is constructed with coverage probability − 3 1 − α + O(n 2 ).

Our contributions in this paper are twofold. Firstly, we develop two estimative prediction intervals from normal random variables with parameter (µ, σ2). We have these two intervals by setting diﬀerent estimators for µ and σ2. Then, improved prediction intervals are constructed and assessment of such intervals are carried out via their expected length. Note that our parametric estimative prediction intervals setting is diﬀerent to that of Kabaila and Syuhada (2007) who considered parametric versus nonparametric prediction intervals. The second contribution lies in our attempt to assess an estimative prediction interval by calculating O(n−1) terms of coverage probability and expected length together an set to be minimum to reach optimal 1 − α. In fact, we propose that α may be considered as parameter instead of given value. The O(n−1) terms then depend on α and parameter vector (µ, σ2).

Whilst improved prediction interval may correct the coverage property of estimative one, in fact we may not be able to ﬁnd optimal coverage probability by incorporating O(n−1) term in the asymptotic expected length. This is due − 3 to the absorbtion of unexpected coverage and replaced by O(n 2 ).

The remainder of this paper is organized as follows. In Section 2, a de- scription of estimative and improved prediction intervals is presented along with the expected length of these prediction intervals. We also introduce the total unexpected coverage and length. We provide an example of independent observations from normal distribution and this is given in Section 3. A numerical analysis is presented in Section 4 aimed to illustrate diﬀerent estimative prediction intervals. Section 5 explores the optimal coverage probability. PI for independent observations 4933

2 Prediction Intervals: Unexpected Coverage And Length

A general formulation of estimative prediction interval along with its expected length is described briefly in this Section. Suppose that the available i.i.d. data are Y1,...,Yn from the model with parameter vector θ. For unknown θ, the estimative prediction interval for future observation Z = Yn+1 is I Θb = h i L(Θ)b ,U(Θ)b for a specified estimator Θ.b The coverage probability P Z ∈ −1 I Θb differs from 1 − α by c(θ) n . Meanwhile, its expected length is of the form p(θ) + q(θ) n−1 + ··· , where p(θ) > 0. We propose that, for our objective, the O(n−1) terms in the asymptotic expansion of coverage probability and of expected length will depend on α. Thus, the O(n−1) terms for coverage probability and expected lengths are c(α, θ) and q(α, θ), respectively. It is clear that if c(α, θ) = 0 then we have an improved prediction interval. Otherwise, we seek to find an optimal 1 − α by having minimum c(α, θ) and so does q(α, θ). The total unexpected coverage and length is c(α, θ) + q(α, θ). To construct an improved prediction interval, first we note that d(Θ)b = −c(Θ)b n−1/ 2 f(z; θb) where z is estimative prediction limit whilst f(z; θˆ) h is its pdf. The relevant improved prediction interval is I+(Θ)b = L(Θ)b − i d Θb ,U(Θ)b + d Θb in which its coverage probability now is bounded to O(n−3/2). The corresponding expected length for improved prediction interval is of the form p(α, θ) + q+(α, θ) n−1 + ··· , from which the total unexpected coverage and length is k(α, θ) = q+(α, θ).

3 The Case Of Normal Random Variables

In what follows, we demonstrate the derivation of estimative prediction interval and its expected length for the case of normal random variables. An improved prediction interval is also constructed. Suppose that Y1,...,Yn are i.i.d. normally distributed with unknown mean µ and unknown variance σ2. We are concerned with a prediction interval for Z = Yn+1 which is independent and from the same distribution. Let θ = (µ, σ2). For a speciﬁed estimator 2 2 Θb = (µ,b σb ) of (µ, σ ), the 1 − α estimative prediction interval is h i I(Θ) = µ − z1− α σ , µ + z1− α σ b b 2 b b 2 b 4934 Khreshna Syuhada and Rizky Saputra

and its coverage probability is Pθ Z ∈ I(Θ)b which is equal to

(µ − µ) + z1− α σ (µ − µ) − z1− α σ E Φ b 2 b − Φ b 2 b = 1−α+c(θ) n−1 +··· θ σ σ

−1 2 −1 2 2 −1 2 where c(θ) n = −(σ ) z1− α φ z1− α E (µ−µ) +(σ ) z1− α φ z1− α E σ − 2 2 b 2 2 b 2 1 2 −2 3 2 2 2 σ − (σ ) z1− α + z α φ z1− α E (σ −σ ) by using a similar argu- 4 2 1− 2 2 b ment to that used in Syuhada (2008, p.17). As explained before, the O(n−1) arises due to parameter variability. The expected length of I(Θ)b is

2 1 E length of I(Θ) = 2 z1− α E (σ ) 2 . (1) b 2 b Now, by Taylor expansion

2 2 1 2 1 1 2 − 1 2 2 1 2 − 3 2 2 (σ ) 2 = (σ ) 2 + (σ ) 2 σ − σ − (σ ) 2 σ − σ + ··· b b 2 2 b b 2 2 b 2 σb =σ 8 σb =σ

2 1 2 − 1 2 2 1 2 − 3 2 We obtain (1) = 2 z1− α (σ ) 2 +z1− α (σ ) 2 E σ −σ − z1− α (σ ) 2 E (σ − 2 2 b 4 2 b 2 2 2 1 −1 σ ) + ··· which is equal to 2 z α (σ ) 2 + q(θ) n + ··· for the usual sorts 1− 2 2 2 of estimators σb of σ , where q(θ) is determined by the asymptotic moments 2 of σb . An improved prediction interval is constructed by absorbing the O(n−1) term. The resulting interval is

+ h i I (Θ) = µ − z1− α σ − d(Θ) , µ + z1− α σ + d(Θ) b b 2 b b b 2 b b

−1 2 − 1 where d(Θ) = −c(Θ) n /(2 (σ ) 2 φ z1− α ). The coverage probability of this b b b 2 improved prediction interval is 1 − α + O(n−3/2) whilst the expected length of this interval is

+ 2 1 E length of I (Θ) = 2 z1− α E (σ ) 2 + 2 E d Θ (2) b 2 b b

2 1 By, again, the Taylor expansion for (σ ) 2 above and the following Tay- b ∂ d Θ ∂ d Θ b b 2 2 lor expansion d Θb = d(θ) + ∂ µ µ − µ + ∂ σ2 σ − σ + b Θ=b θ b b Θ=b θ b 2 1 2 − 1 2 ··· , we ﬁnd that (2) is equal to 2 z1− α (σ ) 2 + (σ ) 2 z1− α E (µ − µ) + 2 2 b 1 2 − 3 3 2 2 2 2 1 + −1 + (σ ) 2 z α E (σ − σ ) = 2 z1− α (σ ) 2 + q (θ) n + ··· , where q (θ) is 4 1− 2 b 2 2 determined by the asymptotic moments of σb and µb. It is important to note here that the expected length of the improved prediction interval does not depend on the bias of parameter estimators (see Kabaila and Syuhada, 2010). PI for independent observations 4935

4 Numerical Analysis

We illustrate the derivation of estimative and improved prediction intervals along with their expected lengths as follows. The ﬁrst intervals, say I1 Θb + and I1 Θb , are developed using the following estimator. Observe that the log- n 2 likelihood function (to within an additive constant) is `(θ; y) = − 2 log σ − 1 Pn 2 2 2 σ2 t=1 (yt − µ) . The maximum likelihood estimator for θ = (µ, σ ) is Θb = (µ, σ2), where µ = 1 Pn Y , which is unbiased, and σ2 = 1 Pn Y − µ2. b b b n t=1 t b n t=1 t b Using a method very similar to that used by Vidoni (2004, p.144), we de- rive the expected information matrix and the conditional bias as follows. Let 2 2 θ = (θ1, θ2) = (µ, σ ) and Θb = (Θb 1, Θb 2) = (µ,b σb ) is the corresponding ML estimator. The partial derivatives of the log-likelihood function with respect to the 2 −1 1 2 −2 2 −3 Pn parameters are `(θ; y)11 = −(σ ) n; `(θ; y)22 = 2 (σ ) n−(σ ) t=1 (yt − 2 2 −2 Pn µ) ; `(θ; y)12 = −(σ ) t=1 (yt − µ), where, for example, `(θ; y)12 denotes 2 n ∂ `(θ; y)/(∂ θ1 ∂ θ2). Thus, whilst i12 = 0, i11 = E − `(θ; y)11 = σ2 ; i22 = n E − `(θ; y)22 = 2 (σ2)2 . The asymptotic bias for µ and σ2 are, respectively, b(θ) = − 1 λ + b b 1 2 1,11 11 11 1 11 22 1 22 11 λ1,1,1 i i − 2 λ1,22 + λ1,2,2 i i and b(θ)2 = − 2 λ2,11 + λ2,1,1 i i + 22 22 λ2,22 + λ2,2,2 i i , which are equal to (Barndorﬀ-Nielsen and Cox, 1994, 1 11 11 1 11 22 p.185) b(θ)1 = λ1,11 + 2 λ111 i i + λ2,12 + 2 λ122 i i and b(θ)2 = 1 22 11 1 22 22 λ1,21 + λ211 i i + λ2,22 + λ222 i i , where, for s, t, u = 1, 2, λt,su = 2 2 E `(θ; y)t `(θ; y)su and λsut = E `(θ; y)sut .

We ﬁnd that b(θ)1 = 0 which agrees to the fact that µb is unbiased es- 2 timator. To obtain the asymptotic bias of σb , we use the following results 2 −1 λ1,21, λ2,22, λ222. We obtain b(θ)2 = −σ n . (µ−µ)+z α σ (µ−µ)−z α σ b 1− 2 b b 1− 2 b Now, to obtain d(θ) via c(θ), let G Θ;b θ = Φ σ −Φ σ and the partial derivatives of G Θ;b θ with respect to the parameter estimators ∂ G Θ;b θ ∂2 G Θ;b θ ∂ G Θ;b θ 2 −1 α α are = 0; 2 = −2 (σ ) z1− φ z1− ; 2 = ∂ µb ∂ µb 2 2 ∂ σb Θ=b θ Θ=b θ Θ=b θ ∂2 G Θ;b θ 1 2 −1 α α 2 −2 α 3 α (σ ) z1− φ z1− ; 2 2 = − (σ ) z1− +z1− α φ z1− . Thus, 2 2 ∂ (σb ) 2 2 2 2 Θ=b θ −1 2 −1 2 −1 1 2 −1 2 −1 we ﬁnd c(θ) n = (σ ) z α φ z α (−σ ) n + −2 (σ ) z α φ z α σ n + 1− 2 1− 2 2 1− 2 1− 2 1 1 2 −2 3 2 2 −1 − (σ ) z1− α + z α φ z1− α 2 (σ ) n which is equal to 2 2 2 1− 2 2 1 3 −1 −2 z1− α φ z1− α − z1− α + z α φ z1− α n . Note that c(θ) does not 2 2 2 2 1− 2 2 depend on the parameter vector θ = (µ, σ2); c(θ) → 0 as α → 0. 4936 Khreshna Syuhada and Rizky Saputra

Table 1: The coverage probabilities (estimated) of the estimative and improved 0.95 prediction interval. Standard errors are in brackets. n Prediction interval m = 200 m = 500 50 Estimative 0.9441(0.0009) 0.9445(0.0006) Improved 0.9522(0.00002) 0.9522(0.000006) 100 Estimative 0.9473(0.0006) 0.9477(0.0004) Improved 0.9511(0.000008) 0.9511(0.000003) 200 Estimative 0.9490(0.0004) 0.9487(0.0003) Improved 0.9506(0.000004) 0.9506(0.000002)

The expected length of I1(Θ)b is E length of I1(Θ)b which is equal to 2 1 2 − 1 2 2 1 2 − 3 2 2 2 2 z1− α (σ ) 2 + z1− α (σ ) 2 E σ − σ − z1− α (σ ) 2 E (σ − σ ) + ··· = 2 2 b 4 2 b 2 1 3 2 1 −1 −1 2 z α (σ ) 2 − z α (σ ) 2 n + ··· , in which we can observe that the O(n ) 1− 2 2 1− 2 term above depend on parameter θ; although its expected length does not depend on the asymptotic estimator bias. −1 2 − 1 1 2 2 1 −1 Furthermore, d(θ) = −c(θ) n / 2 (σ ) 2 φ z1− α = 5+ z α z1− α (σ ) 2 n . 2 4 1− 2 2 As shown before (by Taylor expansion), E d(Θ)b = E d(θ) + ··· . Conse- 1 2 2 1 −1 quently, E(d(Θ))b = 5 + z α z1− α (σ ) 2 n + ··· . The expected length of 4 1− 2 2 + + 2 1 improved prediction interval 1, I Θ , is E length of I (Θ) = 2 z α (σ ) 2 + 1 b 1 b 1− 2 2 − 1 2 −1 1 2 − 3 2 2 −1 1 2 2 1 −1 z1− α (σ ) 2 (−σ ) n − z1− α (σ ) 2 2 (σ ) n +2 5+z α z1− α (σ ) 2 n + 2 4 2 4 1− 2 2 2 1 1 2 2 1 −1 ··· = 2 z1− α (σ ) 2 + 2 + z α z1− α (σ ) 2 n + ··· . 2 2 1− 2 2 A Monte Carlo simulation is carried out to estimate the estimative and improved prediction intervals from independent observations. Note that we adopt the method of Kabaila and Syuhada’s (2008, 2010) prediction limit in order to obtain improved prediction interval eﬃciently. We begin by running simulation data from normal distribution with mean µ = 2 and variance σ2 = 1, with sample size n = 50, 100, 200. The replications we have used are m = 200, 500. Table 1 reports the estimative and improved prediction intervals when Max- imum Likelihood estimators are employed. It is shown an improvement, in terms of coverage probability, of the improved prediction interval over the estimative one. The standard errors are reasonably small. Also, as the increase of sample size and number of iteration, the coverage probability is closer to the target value 0.95. All the computations reported are performed with programs written using MATLAB and the MATLAB Statistics toolbox. PI for independent observations 4937

Table 2: The expected length of the estimative and improved 0.95 prediction interval. n Prediction interval m = 200 m = 500 50 Estimative 3.8822 3.8951 Improved 3.9575 3.9581 100 Estimative 3.9098 3.9064 Improved 3.9389 3.9391 200 Estimative 3.9133 3.9186 Improved 3.9294 3.9295

The expected length of prediction intervals are presented in Table 2, where the expected length of estimative prediction interval is shorter than that of improved prediction interval. This corresponds to the coverage probability of these prediction intervals, shown in Table 1, in which the coverage probability of estimative prediction interval is lower than of the improved one. Nonethe- less, the expected length of improved prediction interval is closer to the target 3.92. The improvement is also observed as the sample size and number of iteration increase. Now consider the estimative prediction interval 2, I2 Θb , obtained using the estimator for θ = (µ, σ2) is Θ = (µ, S2) where µ = µ and S2 = n σ2. e e e b n−1 b The estimator S2 is unbiased, i.e. ES2 − σ2 = 0. The variance of S2 is 2 4 −1 −1 1 2 −1 2 −1 V ar(S ) = 2 σ n . We obtain c(θ) n = −2 (σ ) z α φ z α σ n 2 1− 2 1− 2 1 1 2 −2 3 2 2 −1 + − (σ ) z1− α + z α φ z1− α 2 (σ ) n which is equal to 2 2 2 1− 2 2 1 3 −1 −1 −z1− α φ z1− α − z1− α + z α φ z1− α n . As for I1(Θ),b the O(n ) 2 2 2 2 1− 2 2 term in the asymptotic expansion of the coverage probability for I2(Θ)b does not 2 1 depend on parameter vector θ. In fact, E length of I (Θ) = 2 z α (σ ) 2 − 2 b 1− 2 1 2 1 −1 z α (σ ) 2 n + ··· . 2 1− 2 Figure 1 presents the coverage probability of estimative prediction interval 1 and 2 for various n. It is shown that the coverage probability of interval 2 (PI2) is closer to the target value than of interval 1 (PI1). This means that PI2 is more accurate. + The improved prediction interval 2, I2 Θe , and its expected lenth are con- −1 2 − 1 1 structed by ﬁrst calculating d(θ) = −c(θ) n / 2 (σ ) 2 φ z α = 3 + 1− 2 4 2 2 1 −1 z α z1− α (σ ) 2 n . By Taylor expansion, E d(Θ)b = E d(θ) + ··· . Con- 1− 2 2 1 2 2 1 −1 sequently, E(d(Θ))b = 3 + z α z1− α (σ ) 2 n + ··· . The expected length 4 1− 2 2 4938 Khreshna Syuhada and Rizky Saputra

0.99 est PI 1 0.98 est PI 2 1−α 0.97

0.96

0.95

0.94

coverage probability 0.93

0.92

0.91

0.9 0 50 100 150 200 n

Figure 1: Coverage Probability of Estimative PIs

+ + 2 1 1 2 − 3 2 2 −1 of I Θ is E length of I (Θ) = 2 z α (σ ) 2 −2 z α (σ ) 2 2 (σ ) n + 2 e 2 b 1− 2 1− 2 8 1 2 2 1 −1 2 1 1 2 2 1 −1 2 3+z α z1− α (σ ) 2 n +··· = 2 z1− α (σ ) 2 + 2 + z α z1− α (σ ) 2 n + 4 1− 2 2 2 2 1− 2 2 ··· , that has same O(n−1) term as in the expected length of improved prediction interval 1.

Remark 4.1. The standard or classical 1 − α prediction interval for Y is h i √ n+1 ¯ −1 µ − t1− α ,n−1 σ , µ + t1− α ,n−1 σ , where µ = Y, σ = S 1 + n . As stated b 2 b b 2 b b b in Kabaila and Syuhada (2007, p.2675), this interval has asymptotic expected z3 2 1 −1 1− α length 2 z α (σ ) 2 + σ n 2 + z α + ··· in which the coeﬃcient of 1− 2 2 1− 2 O(n−1) term is the same as in the expected length of improved prediction interval 1 and 2 above. We may assess the expected length of a standard prediction interval since its coverage probability is exactly 1 − α. On the other hand, we may not be able to assess the expected length of an estimative prediction interval since it has the same order as its coverage probability; thus, in this case, we may use an improved prediction interval. In other words, diﬀerent order of asymptotic coverage probability and expected length is required.

Remark 4.2. Kabaila and Syuhada (2007, Section 3) compared the standard prediction interval against nonparametric prediction interval. They found the the latter is less eﬃcient than the former, i.e. the nonparametric prediction interval has higher expected length. A parametric prediction interval, with PI for independent observations 4939

0.01

0.005

0 ) σ

−0.005

−0.01 ) and EL−(2*z* α −0.015 CP−(1− −0.02

CovProb −0.025 ExpLength α −0.03 0 0.02 0.04 0.06 0.08 0.1 α

Figure 2: Optimal Coverage Probability of Estimative PI 2 asymptotic coverage probability, may be proposed as an alternative prediction interval to the standard one. The asymptotic expected length of the proposed interval, however, may be the same (but not smaller than) of the standard prediction interval.

5 Optimal Coverage Probability

The assessment of a prediction interval may alternatively be done by consider- ing α as a parameter instead of a given value. In particular, we set the O(n−1) terms in both asymptotic expansion of coverage probability and of expected length depend on α, i.e. c(α, θ) and q(α, θ), respectively. Our aim is to ﬁnd optimal coverage probability 1 − α. 2 1 Deﬁne ` = 2 z α (σ ) 2 , which is in fact the true expected length of pre- α 1− 2 diction interval. The O(n−1) term of having coverage probability for the estimative prediction interval 1, I1 Θb , called unexpected coverage, is c1(α, θ) = 1 2 − 1 3 2 − 1 `α (σ ) 2 −2 z1− α φ z1− α − z1− α +z α φ z1− α which is equal to −`α (σ ) 2 φ − 2 2 2 2 1− 2 2 2 1 3 1 2 − 3 2 − 2 − 1 `α (σ ) 2 `α (σ ) 2 `α (σ ) 2 −1 2 2 + 8 φ 2 whilst the O(n ) term of having asymp- 3 2 1 totic expected length of I Θ or unexpected length is q (α, θ) = − 2 z α (σ ) 2 = 1 b 1 4 1− 2 3 − 4 `α. Total unexpected coverage and length is c(α, θ)+q(α, θ). Therefore, the optimal coverage probability is 0.946 instead of 0.95 which is commonly used. For the second estimative prediction interval, I2 Θb , the unexpected coverage 4940 Khreshna Syuhada and Rizky Saputra

1 1 1 3 1 2 − 2 − 2 − 3 2 − 2 − `α (σ ) 2 `α (σ ) 2 1 `α (σ ) 2 `α (σ ) 2 `α (σ ) 2 and length are − 2 φ 2 − 2 2 + 8 φ 2 1 and − 4 `α, respectively; the optimal coverage probability is 0.977 (see Figure 2).

Acknowledgements. The ﬁrst author is grateful to Assoc. Prof. Paul Kabaila for a thoughtful discussion.

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Received: May 14, 2015; Published: July 17, 2015